nLab
model structure on functors

Idea

The model category structures on functor categories are models for (∞,1)-categories of (∞,1)-functors.

For $C$ a model category and $D$ any small category there are two “obvious” ways to put a model category structure on the functor category $[D,C]$, called the projective and the injective model structures. For completely general $C$, neither one need exist. The projective model structure exists as long as $C$ is cofibrantly generated, while the injective model structure exists as long as $C$ is combinatorial.

A related kind of model structure is the Reedy model structure/generalized Reedy model structure on functor categories, which applies for any model category $C$, but requires $D$ to be a very special sort of category, namely a Reedy category/generalized Reedy category.

In the special case that $C=$ SSet is the standard model category of simplicial sets the projective and injective model structure on the functor categories $[D,\mathrm{SSet}]$ are described in more detail at global model structure on simplicial presheaves.

Definition

Definition

For $C$ a combinatorial model category or, in the projective case, just a cofibrantly generated model category, and $D$ a small category there exist the following two (combinatorial) model category structures on the functor category $[D,C]$:

• the projective structure $[D,C{]}_{\mathrm{proj}}$: weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in $C$.

• the injective structure $[D,C{]}_{\mathrm{inj}}$: weak equivalences and cofibrations are the natural transformations that are objectwise such morphisms in $C$.

More generally, if $C$ is in addition a simplicial model category and $D$ a smooth sSet-enriched category, then the sSet-enriched functor category, also denoted $[D,C]$, carries the above two model strutures.

Properties

In all of the following, let $S$ be an excellent model category. The standard example is the model structure on simplicial sets, ${\mathrm{sSet}}_{\mathrm{Quillen}}$.

Let $D$ (and $D_1$, $D_2$, …) be a combinatorial $S$-enriched model category.

Moreover, $C$ in the following is assumed to be either an ordinary small category, or, more generally, it is a small $S$-enriched category.

If $S=$ sSet${}_{\mathrm{Quillen}}$ and $C$ is an ordinary small category, then then model structures discussed here are instances of the model structure on simplicial presheaves. If $C$ is itself $\mathrm{sSet}$-enriched, then they are instances of the model structure on sSet-enriched presheaves.

General

Proposition

The projective and injective structures $[D,C{]}_{\mathrm{proj}}$ and $[D,C{]}_{\mathrm{inj}}$, def. 1

• are indeed model category structures;

• are themselves combinatorial model categories;

• are right or left proper model categories if $C$ is right or left proper, respectively.

• are $S$-enriched model categories (e.g.simplicial model categories) with respect to the $S$-enrichment for which the $S$-tensoring is objectwise that of $C$.

The existence of the unenriched model structure apears as HTT, prop. 2.8.2 The enriched case is HTT, prop. A.3.3.2 and the remarks following that. The statement about properness appears as HTT, remark A.2.8.4.

Cofibrant generation

Proposition

Let $C$ be an ordinary small category.

The cofibrations in $[C,A{]}_{\mathrm{proj}}$ are generated from (i.e. are the weakly saturated class of morphisms defined by) the morphisms of the form

$${\mathrm{Id}}_{C(c,-)}\cdot i:C(c,-)\cdot a\to C(c,-)\cdot b$$Id_{C(c,-)}\cdot i : C(c,-)\cdot a \to C(c,-) \cdot b

for all $c\in C$ and $i:a\to b$ a generating cofibration in $A$. Here the dot denotes the tensoring of $A$ over sets, i.e. $C(c,-)\cdot a$ is the functor that sends $c\prime \in C$ to the coproduct ${\coprod }_{C(c,c\prime )}A$ of $\mid C(c,c\prime )\mid$ copies of $A$.

In particular, every cofibration if $[C,A{]}_{\mathrm{proj}}$ is in particular a cofibration in $[C,A{]}_{\mathrm{inj}}$. Similarly, every fibration in $[C,A{]}_{\mathrm{inj}}$ is in particular a fibration in $[C,A{]}_{\mathrm{proj}}$

This is argued in the beginning of the proof of HTT, lemma A.2.8.3.

Relation to other model structures

Corollary

The identity functors

$$[D,C{]}_{\mathrm{inj}}\stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\underset{\mathrm{Id}}{\to }}[D,C{]}_{\mathrm{proj}}$$[D,C]_{inj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{proj}

form a Quillen equivalence (with $\mathrm{Id}:[D,C{]}_{\mathrm{proj}}\to [D,C{]}_{\mathrm{inj}}$ being the left Quillen functor).

If $D$ is a Reedy category this factors through the Reedy model structure

$$[D,C{]}_{\mathrm{inj}}\stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\underset{\mathrm{Id}}{\to }}[D,C{]}_{\mathrm{Reedy}}\stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\underset{\mathrm{Id}}{\to }}[D,C{]}_{\mathrm{proj}}$$[D,C]_{inj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{Reedy} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{proj}

Functoriality in domain and codomain

Proposition

The functor model structures depend Quillen-functorially on their codomain, in that for

$$D_1\stackrel{\stackrel{L}{\leftarrow }}{\underset{R}{\to }}{D}_2$$D_1 \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D_2

a $S$-enriched Quillen adjunction between combinatorial $S$-enriched model categories, postcomposition induces $S$-enriched Quillen adjunctions

$$[C,D_1{]}_{\mathrm{proj}}\stackrel{\stackrel{[C,L]}{\leftarrow }}{\underset{[C,R]}{\to }}[C,D_2{]}_{\mathrm{proj}}$$[C,D_1]_{proj} \stackrel{\overset{[C,L]}{\leftarrow}}{\underset{[C,R]}{\to}} [C,D_2]_{proj}

and

$$[C,D_1{]}_{\mathrm{inj}}\stackrel{\stackrel{[C,L]}{\leftarrow }}{\underset{[C,R]}{\to }}[C,D_2{]}_{\mathrm{inj}}.$$[C,D_1]_{inj} \stackrel{\overset{[C,L]}{\leftarrow}}{\underset{[C,R]}{\to}} [C,D_2]_{inj} \,.

Moreover, if $(L⊣R)$ is a Quillen equivalence, then so is $([C,L]⊣[C,R])$.

For the case that $C$ is a small category this is (Lurie, remark A.2.8.6), for the enriched case this is (Lurie, prop. A.3.3.6).

The Quillen-functoriality on the domain is more asymmetric.

Proposition

For $p:C_1\to C_2$ a functor between small categories or an $S$-enriched functor between $S$-enriched categories, let

$$(p_{!}⊣p^{*}⊣p_{*}):[C_2,D]\stackrel{\stackrel{p_{!}}{\to }}{\stackrel{\stackrel{p^{*}}{\to }}{\underset{p_{*}}{\leftarrow }}}[C_1,D]$$(p_! \dashv p^* \dashv p_*) : [C_2,D] \stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^*}{\to}}{\underset{p_*}{\leftarrow}}} [C_1,D]

be the adjoint triple where $p^{*}$ is precomposition with $p$ and where $p_{!}$ and $p_{*}$ are left and right Kan extension along $p$, respectively.

Then we have Quillen adjunctions

$$(p_{!}⊣p^{*}):[C_1,D{]}_{\mathrm{proj}}\stackrel{\stackrel{p_{!}}{\to }}{\underset{p^{*}}{\leftarrow }}[C_2,D{]}_{\mathrm{proj}}$$(p_! \dashv p^*) : [C_1,D]_{proj} \stackrel{\overset{p_!}{\to}}{\underset{p^*}{\leftarrow}} [C_2,D]_{proj}

and

$$(p^{*}⊣p_{*}):[C_1,D{]}_{\mathrm{inj}}\stackrel{\stackrel{p^{*}}{\leftarrow }}{\underset{p_{*}}{\to }}[C_2,D{]}_{\mathrm{inj}}.$$(p^* \dashv p_*) : [C_1,D]_{inj} \stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}} [C_2,D]_{inj} \,.

For $C$ not enriched this appears as (Lurie, prop. A.2.8.7), for the enriched case it appears as (Lurie, prop. A.3.3.7).

Remark

In the $\mathrm{sSet}$-enriched case, if $p:D_1\to D_2$ is an weak equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.

Proposition

For $C$ a combinatorial simplicial model category, the (∞,1)-category presented by $[D,C{]}_{\mathrm{proj}}$ and $[D,C{]}_{\mathrm{inj}}$ under the above assumptions is the (∞,1)-category of (∞,1)-functors $\mathrm{Func}(D,C^{\circ })$ from the ordinary category $D$ to the $(\mathrm{\infty },1)$-category presented by $C$.

See (∞,1)-category of (∞,1)-functors for details.

Relation to homotopy Kan extensions/limits/colimits

Often functors $D\to C$ are thought of as diagrams in the model category $C$, and one is interested in obtaining their homotopy limit or homotopy colimit or, generally, for $p:D\to D\prime$ any functor, their left and right homotopy Kan extension.

These are the left and right derived functors $\mathrm{HoLan}:=𝕃p_1$ and $\mathrm{HoRan}:=ℝp_{*}$ of

$$[D,C{]}_{\mathrm{proj}}\stackrel{p_{!}}{\to }[D\prime ,C{]}_{\mathrm{proj}}$$[D,C]_{proj} \stackrel{p_!}{\to} [D',C]_{proj}

and

$$[D,C{]}_{\mathrm{inj}}\stackrel{p_{*}}{\to }[D\prime ,C{]}_{\mathrm{inj}}$$[D,C]_{inj} \stackrel{p_*}{\to} [D',C]_{inj}

respectively.

For more on this see homotopy Kan extension. For the case that $D\prime =*$ this reduces to homotopy limit and homotopy colimit.

Examples

Examples of cofibrant objects in the projective model structure are discussed at

• projectively cofibrant diagram.

Related concepts

• Reedy model structure, generalized Reedy model structure,

• model structure on homotopical presheaves, model structure on simplicial presheaves

References

The plain situation is the topic of section A.2.8 of

• Jacob Lurie, Higher Topos Theory

The enriched situation is section A.3.3 there.